Key Equations

• Derivative of a vector-valued function
  ${\bf{r}}^{\prime}(t)=\underset{\Delta{t}\to{0}}{\lim}\frac{{\bf{r}}(t+\Delta{t})-{\bf{r}}(t)}{\Delta{t}}$
• Principal unit tangent vector
  ${\bf{T}}(t)=\dfrac{{\bf{r}}^{\prime}(t)}{\parallel{\bf{r}}^{\prime}(t)\parallel}$
• Indefinite integral of a vector-valued function
  $\displaystyle\int [f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}]dt=\left[\displaystyle\int f(t)dt\right]{\bf{i}}+\left[\displaystyle\int g(t)dt\right]{\bf{j}}+\left[\displaystyle\int h(t)dt\right]{\bf{k}}$
• Definite integral of a vector-valued function
  $\displaystyle\int_{a}^{b} [f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}]dt=\left[\displaystyle\int_{a}^{b} f(t)dt\right]{\bf{i}}+\left[\displaystyle\int_{a}^{b} g(t)dt\right]{\bf{j}}+\left[\displaystyle\int_{a}^{b} h(t)dt\right]{\bf{k}}$

Glossary

definite integral of a vector-valued function

the vector obtained by calculating the definite integral of each of the component functions of a given vector-valued function, then using the results as the components of the resulting function

derivative of a vector-valued function

the derivative of a vector-valued function ${\bf{r}}(t)$ is ${\bf{r}}^{\prime}(t)=\underset{\Delta{t}\to{0}}{\lim}\frac{{\bf{r}}(t+\Delta{t})-{\bf{r}}(t)}{\Delta{t}}$, provided the limit exists

indefinite integral of a vector-valued function

a vector-valued function with a derivative that is equal to a given vector-valued function

principal unit tangent vector

a unit vector tangent to a curve $C$

tangent vector

to ${\bf{r}}(t)$ at $t=t_{0}$ any vector ${\bf{v}}$ such that, when the tail of the vector is placed at point ${\bf{r}}(t_{0})$ on the graph, vector ${\bf{v}}$ is tangent to curve $C$